Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces

Abstract

The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.

1. Introduction

In this paper, we study the existence of periodic, almost periodic, and asymptotic almost periodic solutions of the following functional difference equations with infinite delay:

(1.1)

assuming that this system possesses a bounded solution with some property of stability. In (1.1) , and denotes an abstract phase space which we will define later.

The abstract space was introduced by Hale and Kato [1] to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham [2], Hino et al. [3]. The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami [4, 5], Elaydi et al. [6], Cuevas and Pinto [7, 8], Cuevas and Vidal [9], and Cuevas and Del Campo [10].

As usual, we denote by , , and the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let be the -dimensional complex Euclidean space with norm . the set .

If is a function, we define for, the function by , . Furthermore is the function given for , with .

The abstract phase space , which is a subfamily of all functions from into denoted by , is a normed space (with norm denoted by ) and satisfies the following axioms.

• (A) There is a positive constant and nonnegative functions and on with the property that is a function, such that , then for all , the following conditions hold:

  • (i) ,

  • (ii) ,

  • (iii) .

• (B) The space is a Banach space.

  We need the following property on .

• (C) The inclusion map is continuous, that is, there is a constant , such that , for all , where represents the bounded functions from into .

Axiom (C) says that any element of the Banach space of the bounded functions equipped with the supremum norm is on .

Remark 1.1.

Using analogous ideas to the ones of [3], it is not difficult to prove that Axiom (C) is equivalente to the following.

• (C') If a uniformly bounded sequence in converges to a function compactly on (i.e., converges on any compact discrete interval in ) in the compact-open topology, then belong to and as .

Remark 1.2.

We will denote by (, and ) or simply by , the solution of (1.1) passing through , that is, , and the functional equation (1.1) is satisfied.

During this paper we will assume that the sequences and are bounded. The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. In Section 4 we analyze separately the cases where is periodic and when it is almost periodic. Thus, in Section 4.1 assuming that the system (1.1) is periodic and the existence of a bounded solution (particular solution) which is uniformly stable and the phase space satisfies only the axioms (A)–(C), we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, in Section 4.2 considering that system (1.1) is almost periodic and the existence of a bounded solution and whenever the phase space satisfies the axioms (A)–(C), but here it is also necessary that verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system (1.1) has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.

In [11, 12] the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in [13] the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya's because, firstly, in our work we consider both cases, namely, when is periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of (1.1) with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.

Furthermore, we cite the articles [14–16] which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.

2. Fading Memory Spaces and Implications

Following the terminology given in [3], we introduce the family of operators on , , as

(2.1)

with . They constitute a family of linear operators on having the semigroup property for . Immediately, the following result holds from Axiom (A):

(2.2)

Now, given any function such that , we have the following decomposition:

(2.3)

where

(2.4)

Then, we have the following decomposition of , for , where

(2.5)

and for all . Note that

(2.6)

Let

(2.7)

be a subset of , and let be the restriction of to . Clearly, the family , , is also a strongly continuous semigroup of bounded linear operators on . It is given explicitly by

(2.8)

for .

Definition 2.1.

A phase space that satisfies axioms (A)-(B) and () or () and such that the semigroup is strongly stable is called a fading memory space.

Remark 2.2.

Remember that a strongly continuous semigroup is strongly stable if for all , as .

Thus, we have the following result.

Lemma 2.3.

Let , with , where is a fading memory space. If as , then as .

Proof.

Firstly, we note that as before, , where , for and

(2.9)

Then, by definition as because . On the other hand, by hypothesis, as , so it follows from Axiom (C') that . Therefore, we conclude that as .

3. Notations and Preliminary Results

In this section, we review the definitions of (uniformly) almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12].

Definition 3.1.

A sequence is called an almost periodic sequence if the -translation set of ,

(3.1)

is a relatively dense set in for all ; that is, for any given , there exists an integer such that each discrete interval of length contains such that

(3.2)

is called the -translation number of . We will denote by the set of all such sequences. We will write that is a.p. if .

Definition 3.2.

A sequence is called an asymptotically almost periodic sequence if

(3.3)

where is an almost periodic sequence, and as . We will denote by the set of all such sequences. We will write that is a.a.p. if .

In general, we will consider a Banach space.

Definition 3.3.

A function or sequence is said to be almost periodic (abbreviated a.p.) in if for every there is such that among consecutive integers there is one; call it , such that

(3.4)

Denote by all such sequences, and is said to be an almost periodic (a.p.) in .

Definition 3.4.

A sequence , (or ), , equivalently, a function (or, ) is called asymptotically almost periodic if , where and (or, ) satisfying as (or, ). Denote by (or all such sequences, and is said to be an asymptotically almost periodic on (or on ) (a.a.p.) in .

Remark 3.5.

Almost periodic sequences can be also defined for any sequence () or by requiring that consecutive integers are in .

Definition 3.6.

Let . is said to be almost periodic in uniformly for , if for any and every compact , there exists a positive integer such that any interval of length (i.e., among consecutive integers) contains an integer (or equivalently, there is one); call it , for which

(3.5)

is called the -translation number of . We will denote by the set of all such sequences. In brief we will write that is u.a.p. if .

Definition 3.7.

The hull of , denoted by , is defined by

(3.6)

for some sequence , where is any compact set in .

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12]. With the objective to make this manuscript self contained we decided to include the majority of the proofs.

Lemma 3.8.

• (a) If is an a.p. sequence, then there exists an almost periodic function such that for .

• (b) If is an a.p. function, then is an a.p. sequence.

Lemma 3.9. (a) If is an a.p. sequence, then is bounded.

(b) is an a.p. sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limits sequence is also an almost periodic sequence.

(c) is an a.p. sequence if and only if for any sequence of integers , there exist subsequences , such that

(3.7)

where for .

(d), (or, ) is an a.a.p. sequence if and only if for any sequence (or, ) such that and as (or, as ), there exists a subsequence such that converges uniformly on (or ) as .

Lemma 3.10.

Let be an a.a.p. periodic sequence. Then its decomposition,

(3.8)

where is an a.p. sequence while as , is unique.

Lemma 3.11.

Let be almost periodic in uniformly for and continuous in . Then is bounded and uniformly continuous on for any compact set in .

Lemma 3.12.

Let be the same as in the previous lemma. Then, for any sequence , there exist a subsequence of and a function continuous in such that uniformly on as , where is any compact set in . Moreover, is also almost periodic in uniformly for .

Lemma 3.13.

Let be the same as in the previous lemma. Then, there exists a sequence , as such that uniformly on as , where is any compact set in .

Lemma 3.14.

Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in . Then is almost periodic in .

Lemma 3.15.

Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in and for . Then is almost periodic in .

Remark 3.16.

If is a.a.p., then the decomposition , in the definition of an a.a.p. function, is unique (see [18]).

4. Existence of Almost Periodic Solutions

From now on we will assume that the system (1.1) has a unique solution for a given initial condition on and without loss of generality , thus .

We will make the following assumptions on (1.1).

• (H1) is continuous in the second variable for any fixed .

• (H2) System (1.1) has a bounded solution , passing through , , that is, .

For this bounded solution , there is an such that for all . So, we will have to assume that for all , and . Next, we will point out the definitions of stability for functional difference equations adapting it from the continuous case according to Hino et al. in [3].

Definition 4.1.

A bounded solution of (1.1) is said to be:

• (i) stable, if for any and any integer , there is such that implies that for all , where is any solution of (1.1);

• (ii) uniformly stable, abbreviated as "'', if for any and any integer , there is ( does not depend on ) such that implies that for all , where is any solution of (1.1);

• (iii) uniformly asymptotically stable, abbreviated as "'', if it is uniformly stable and there is such that for any , there is a positive integer such that if and , then for all , where is any solution of (1.1);

• (iv) globally uniformly asymptotically stable, abbreviated as "'', if it is uniformly stable and as , whenever is any solution of (1.1).

Remark 4.2.

It is easy to see that an equivalent definition for , being , is the following:

• (iii) is , if it is uniformly stable, and there exists such that if and , then as , where is any solution of (1.1).

4.1. The Periodic Case

Here, we will assume what follows.

• (H3) The function in (1.1) is periodic in , that is, there exists a positive integer such that for all .

Moreover, we will assume what follows.

• () The sequences and in Axiom (A)(iii) are bounded by and , respectively and .

Lemma 4.3.

Suppose that condition () holds. If is a bounded solution of (1.1) such that , then is also bounded in .

Proof.

Let us say that for all . Then by Axiom (A)(iii) and hypothesis () we have

(4.1)

Lemma 4.4.

Suppose that condition () holds. Let be a sequence in such that for all . Assume that as for every and , then in as for each . In particular, if as uniformly in , then in as uniformly in .

Proof.

By Axiom (A)(iii) and hypotheses we have that

(4.2)

In the particular case we obtain

(4.3)

and so as . On the other hand, since is fixed, it follows that

(4.4)

for each . Therefore, we have concluded the proof.

Theorem 4.5.

Suppose that condition () and (H1)–(H3) hold. If the bounded solution of (1.1) is , then is an a.a.p. sequence in , equivalently, (1.1) has an a.a.p. solution.

Proof.

By Lemma 4.3 there exists such that for all , and a bounded (or compact) set such that for all . Let be any integer sequence such that and as . For each , there exists a nonnegative integer such that . Set . Then for all . Since is a bounded set, we can assume that, taking a subsequence if necessary, for all , where . Now, set . Thus,

(4.5)

which implies that is a solution of the system,

(4.6)

through . It is clear that if is , then is also with the same pair as the one for .

Since is bounded for all and , we can use the diagonal method to get a subsequence of such that converges for each as . Thus, we can assume that the sequence converges for each as . Since , by Lemma 4.4 it follows that is also convergent for each . In particular, for any there exists a positive integer such that if ( is the constant given in Axiom A(ii), then

(4.7)

where is the number given by the uniform stability of . Since , it follows from Definition 4.1 and (4.7) that

(4.8)

and by Axiom A(ii) it follows that

(4.9)

This implies that for any positive integer sequence , as , there is a subsequence of for which converges uniformly on as . Thus, the conclusion of the theorem follows from Lemma 3.9(d).

Before proving our following result we remark that if is a.a.p. then there are unique sequences such that , with a.p. and as as . By Lemma 3.9(a) it follows that is bounded and thus . Hence, by Axiom (C) we must have that for all . In particular, for all .

Theorem 4.6.

Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then system (1.1) has an a.p. solution, which is also .

Proof.

It follows from Theorem 4.5 that is an a.a.p. Set (), where is a.p. sequence and as . For the positive integer sequence , by Lemma 3.9(b)–(d) and arguments of the previous theorem, we can choice a subsequence of such that converges uniformly in and uniformly on as and is also a.p. Then, uniformly in , and thus by Lemma 4.4 uniformly in on as and . Since

(4.10)

as , we have for , that is, the system (1.1) has an almost periodic solution, and so we have proved the first statement of the theorem.

In order to prove the second affirmation, notice that since . For any , let be a solution of (1.1) such that and . Again, by Lemma 4.4 as for each , so there is a positive integer such that if , then

(4.11)

Thus, for , we have

(4.12)

Then,

(4.13)

Therefore, there is such that if , then

(4.14)

and hence, for all , where is a pair for the uniform stability of . This shows that if , then

(4.15)

for all , which implies that for all if because is arbitrary. This proves that is .

In the case when we have an asymptotically stable solution of (1.1) we obtain the following result.

Theorem 4.7.

Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period for some positive integer , which is also .

Proof.

Set , . By the proof of Theorem 4.5, there is a subsequence which converges to a solution of (4.6) for each and hence by Lemma 4.4, as . Thus, there is a positive integer such that (), where is obtained from the uniformly asymptotic stability of . Let , and notice that is a solution of (1.1). Since for , that is, , we have

(4.16)

and hence,

(4.17)

because is (see also Remark 4.2). On the other hand, is a.a.p. by Theorem 4.5, then

(4.18)

where is a.p. and as . It follows from (4.17) and (4.18) that

(4.19)

which implies that for all because is a.p.

For the integer sequence , , we have . Then uniformly for all as , and again by Lemma 4.4, uniformly in as . Since , we have for , which implies that (1.1) has a periodic solution of period .

Now, we will proceed to prove that by the use of definition in Remark 4.2. Notice that since then is a solution of (1.1) with the same as the one for . Let be any solution of (1.1) such that . Set . Again, for sufficient large , we have the similar relations (4.12) and (4.14) with and . Thus,

(4.20)

as if , because , , and satisfy (1.1). This completes the proof.

Finally, if the particular solution is , we will prove that system (1.1) has a periodic solution.

Theorem 4.8.

Suppose that and (H1)–(H3) hold and that the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period .

Proof.

By Theorem 4.5, is a.a.p. Then ), where () is an a.p. sequence and as . Notice that is also a solution of (1.1) satisfying . Since is , we have that as , which implies that for all . Using same technique as in the proof of Theorem 4.7, we can show that is a -periodic solution of (1.1).

4.2. The Almost Periodic Case

Here, we will assume that

• (H4) the function in (1.1) is almost periodic in uniformly in the second variable.

By we denote the uniform closure of , that is, . Note that by Lemma 3.12 and by Lemma 3.13.

Lemma 4.9.

Suppose that Axiom (C) is true, and that is an a.p. sequence with , then is a.p.

Proof.

We know that, given , there exists an integer such that each discrete interval of length contains a such that

(4.21)

By Axiom (C) we have

(4.22)

Lemma 4.10.

Suppose that is a fading memory space and is a.a.p. with , then is a.a.p.

Proof.

Since is a.a.p. there are unique sequences and such that is a.p. and as . Then by Lemma 4.9 it follows that is a.p., and by Lemma 2.3 it follows that as . Therefore, is a.a.p.

Theorem 4.11.

Suppose that conditions , (H1)-(H2), and (H4) hold and that is a fading memory space. If the bounded solution of (1.1) is an a.a.p. sequence, then the system (1.1) has an a.p. solution.

Proof.

Since the solution is a.a.p., it follows from Lemma 3.10 that has a unique decomposition , where is a.p. and as . Notice that is bounded. By Lemma 4.3 there is a compact set in such that for all . By Lemma 3.13, there is an integer sequence , , such that as and uniformly on as . Taking a subsequence if necessary, we can also assume that uniformly on , and by Lemma 3.9(b) we have that is also an a.p. sequence. For any , there is a positive integer such that if , then . In this case, we see that uniformly for all as , and hence by Lemma 4.4 in in as . Since

(4.23)

and from the previous considerations the first term of the right-hand side of (4.23) tends to zero as and since as , we have that for all , which implies that (1.1) has an a.p. solution passing through , where for .

We are now in a position to prove the following result.

Theorem 4.12.

Suppose that the assumptions , (H1), (H2), and (H4) hold, and that is a fading memory space. If the bounded solution of (1.1) is , then is a.a.p. Consequently, (1.1) has an a.p. solution which is .

Proof.

Let the bounded solution of (1.1) be with the triple . Let be any positive integer such that as . Set . As previously is a solution of

(4.24)

and is with the same triple . By Lemma A.2, for the set and any there exists such that and for some implies that for all , where is a bounded solution of

(4.25)

passing through and for . Since is uniformly bounded for all and , taking a subsequence if necessary, we can assume that is convergent for each and uniformly on , for some a.p. function . In this case, by Lemma 4.4 there is a positive integer such that if , then

(4.26)

On the other hand, for is a solution of (4.25) with , that is,

(4.27)

where is defined by the relation

(4.28)

To apply Lemma A.2 to (4.24) and its associated equation (4.27), we will point out some properties of the sequence . Since uniformly on , for the above , there is a positive integer such that if , then

(4.29)

which implies that for all . Applying Lemma A.2 to (4.24) and its associated equation (4.27) with the above arguments and condition (4.26), we conclude that for any positive integer sequence , as , and , there is a positive integer such that

(4.30)

and hence by Axiom A(ii) for all if . This implies that the bounded solution of (1.1) is a.a.p. by Lemma 3.9(d). Furthermore, (1.1) has an a.p. solution, which is by Theorem 4.11. This ends the proof.

Appendix

The proof of the following lemmas used ideas developed by Hino et al. in [3] for the functional differential equations with infinite delay and by Song [12] for functional difference equations with finite delay.Lemma A.1. Suppose that , (H1), (H2), and (H4) hold and that is a fading memory space. Let be the bounded solution of (1.1). Let be a positive integer sequence such that , , and uniformly on as , where is any compact subset in and . If the bounded solution is , then the solution of

(A.1)

through , is . In addition, if is , then is also .Proof. Set . It is easy to see that is a solution of

(A.2)

passing though and for all . Since is , then is also with the same pair as the one for . Taking a subsequence if necessary, we can assume that converges to a vector for each as . From (4.23) with , we can see that is the unique solution of (A.1), satisfying because .To show that the solution of (A.1) is , we need to prove that for any and any integer , there exists such that implies that for all , where is a solution of (A.1) with .We know from Lemma 4.4 that as for each ; thus, for any given , if is sufficiently large; say , we have

(A.3)

where comes from the uniform stability of . Let be such that

(A.4)

and let be the solution of (1.1) such that . Then is a solution of (A.2) with . Since is and for , we have

(A.5)

It follows from (A.5) that

(A.6)

Then there exists a number such that for all and , which implies that there is a subsequence of for each , denoted by again, such that for each , and hence by Lemma 4.4 for all as . Clearly, , and the set is compact set . Since is almost periodic in uniformly for , we can assume that, taking a subsequence if necessary, uniformly on as . Taking in , we have , namely, is the unique solution of (A.1), passing through with . On the other hand, for any integer , there exists such that if , then

(A.7)

From (A.5) and (A.7), we obtain

(A.8)

Since is arbitrary, we have for all if and , which implies that the solution of (A.1) is .Now, we consider the case where is . Then the solution of (A.2) is also with the same pair as the one for . Let be the pair for uniform stability of .For any given , if is sufficiently large; say , we have

(A.9)

where is the one for uniformly asymptotic stability of . Let such that , and let , for each fixed , be the solution of (1.1) such that . Then is a solution of (A.2) with . Since is and for each fixed , we have

(A.10)

By the same argument as above, there is a subsequence of , which we will continue calling , such that converges to the solution of (A.1) through and uniformly on as , where is a compact set in with for all and . Then is the unique solution of (A.1), passing through with . On the other hand, by Lemma 4.4 for any integer there exists such that if , then

(A.11)

and hence for . Since is arbitrary, we have

(A.12)

if and ; thus, and the proof is complete.Now, we need to prove the following important lemma. Lemma A.2. Suppose that the assumptions , (H1), (H2), and (H4) hold, that is a fading memory space, that the bounded solution of (1.1) is , and that for each , the solution of (A.1) is unique for any given initial data. Let be a given compact set in . Then for any , there exists such that if , , and is a sequence with for , one has for all , where is any bounded solution of the system

(A.13)

passing through and such that for all .Proof. Suppose that the bounded solution of (1.1) is with the triple . The proof will be by contradiction, we assume that Lemma A.2 is not true. Then for some compact set , there exist , , sequences , , mapping sequences , , and

(A.14)

for sufficiently large , where is a solution of

(A.15)

passing through such that for all and . Since is a bounded subset of , it follows that and are uniformly bounded for all and . We first consider the case where contains an unbounded subsequence. Set . Taking a subsequence if necessary, we may assume from Lemmas 3.12 and 3.9(b) that there is such that uniformly on , , and for as , where are some bounded functions. Since

(A.16)

passing to the limit as , by the similar arguments in the proof of Theorem 4.11, we conclude that is the solution of the following equation:

(A.17)

Similarly, is also a solution of (A.17). By Lemma 4.4 and in as ; it follows from (A.14) that . Notice that is a solution of (A.17), passing through , and is by Lemma A.1. We have . On the other hand, since

(A.18)

as for each , it follows from (A.14) that

(A.19)

This is a contradiction. Thus, the sequence must be bounded. Taking a subsequence if necessary, we can assume that . Moreover, we may assume that and for each , and uniformly on , for some functions , on , and . Since and in as , we have by (A.14), and hence , that is, for all . Moreover, and satisfy the same relation:

(A.20)

The uniqueness of the solutions for the initial value problems implies that for , and hence . On the other hand, and again from Lemma 4.4, and in as , then from (A.14) we have

(A.21)

This is a contradiction, that proves Lemma A.2.